The illustrated zoo of order-preserving functions

Transcription

1 The illustrated zoo of order-preserving functions David Wilding, February Posets (partially ordered sets) underlie much of mathematics, but we often don t give them a second thought. I will describe some of the interesting posets and, more importantly, order-preserving functions between posets that turn up in various branches of mathematics. The order-preserving functions on show will range all the way from the very basic (monotone functions) to the most powerful (residuated functions and order isomorphisms). Contents 1 Introduction 1 2 Monotone functions and order isomorphisms 2 3 Kernels and order embeddings 3 4 Residuated functions 4 5 Conclusion 6 References 6 1 Introduction The notion of a partial order (a reflexive, anti-symmetric and transitive binary relation) goes back at least as far as Leibniz in or around Of course, Leibniz did not give the modern formulation, but a careful reading of his work reveals the key ideas, such as transitivity. In the same way, all men are contained in all animals, and all animals in all corporeal substances; therefore all men are contained in corporeal substances. (Leibniz [2]) 1

2 In fact, Leibniz essentially defined what is now called a join-semilattice. Definition 1 A poset X is a join-semilattice if every pair of elements has a least upper bound in X. That is, for all x 1, x 2 X there should be some element x 1 x 2 X (called the join of x 1 and x 2 ) with x 1, x 2 x 1 x 2 and x 1 x 2 x for all x X satisfying x 1, x 2 x. If every pair of elements in a poset X has a greatest lower bound (denoted x 1 x 2 for x 1, x 2 X and defined dually to least upper bound) then X is called a meet-semilattice, and if X is both a join- and a meet-semilattice then it is simply called a lattice. The most familiar lattice is probably the powerset of a set. In this lattice the partial order is subset inclusion, the join of a pair of subsets is their union and the meet of a pair of subsets is their intersection. It is common to draw posets as elements connected by lines, with the convention that an element x 2 is (strictly) greater than an element x 1 in the partial order if and only there is an upwards path from x 1 to x 2. For example, the powerset of {0, 1, 2} is shown in this way in Figure 1. Such a representation of a poset is called a Hasse diagram. {0, 1, 2} {0, 1} {0, 2} {1, 2} {0} {1} {2} Figure 1: The powerset of {0, 1, 2}. 2 Monotone functions and order isomorphisms Posets are sets with a single binary relation (the order), so it makes sense to ask that functions between posets respect this relation. In other words, we should like to work in the category Pos whose objects are posets and whose morphisms are monotone functions between posets. 2

3 Definition 2 Let X and Y be posets. A function f : X Y is monotone if x 1 x 2 f(x 1 ) f(x 2 ) (1) for all x 1, x 2 X. A monotone function between posets need not respect any additional structure (such as joins and/or meets) that the posets may have. An order isomorphism, on the other hand, respects all structure; the existence of an order isomorphism between posets means that they are actually the same poset. Definition 3 Let X and Y be posets. A function f : X Y is an order isomorphism if f is a monotone bijection and f 1 is monotone. The condition that f 1 be monotone in Definition 3 is required because the inverse of a monotone bijection is not automatically monotone. Indeed, Figure 2 shows a monotone bijection whose inverse is not monotone. Figure 2: A monotone bijection between two 3-element posets. 3 Kernels and order embeddings Any subset of a poset is again a poset, so in particular the image of a monotone function is a poset. The kernel of a monotone function f : X Y is not a poset, however, but is instead an equivalence relation on X. Definition 4 Let f : X Y be monotone. The kernel of f is the equivalence relation ker(f) X X defined by (x 1, x 2 ) ker(f) f(x 1 ) = f(x 2 ) (2) for all x 1, x 2 X. 3

4 If f : X Y is monotone then the set X/ ker(f) of equivalence classes can be partially ordered by setting [x 1 ] [x 2 ] if and only if f(x 1 ) f(x 2 ). This is well-defined by (2). The injective function g : X/ ker(f) Y given by g ( [x] ) = f(x) then satisfies [x 1 ] [x 2 ] g ( [x 1 ] ) g ( [x 2 ] ) (3) for all [x 1 ], [x 2 ] X/ ker(f), so is (in particular) monotone. Moreover, the inverse of the bijection g : X/ ker(f) im(f) is also monotone, so g : X/ ker(f) im(f) is an order isomorphism. This proves the following first isomorphism theorem for posets. Theorem 5 If f : X Y is monotone then X/ ker(f) and im(f) are order isomorphic. Condition (3) motivates the definition of another type of order-preserving function. Definition 6 Let X and Y be posets. A function f : X Y is an order embedding if x 1 x 2 f(x 1 ) f(x 2 ) (4) for all x 1, x 2 X. Order embeddings are, by definition, monotone. It is also clear that they are injective. However, order embeddings are more powerful than monotone injections because the domain and image of an order embedding are order isomorphic, whereas the domain and image of a monotone injection need not be. 4 Residuated functions Residuated functions appear in many familiar and interesting settings because they respect much of a poset s structure, yet do not need to be order isomorphisms. Definition 7 Let X and Y be posets. A function f : X Y is residuated if there is a function f : Y X with f(x) y x f (y) (5) for all x X and all y Y. If f : X Y is residuated then the function f (called the residual of f) is unique and is given by f (y) = max{x X : f(x) y} (6) 4

5 for all y Y. We use the notation f because f sends meets to meets if X and Y are meet-semilattices. Similarly f sends joins to joins if X and Y are join-semilattices. It is an easy exercise to show that every residuated function is monotone, but the converse is not true because (as noted above) monotone functions need not respect joins. Theorem 8 If f : X Y is a residuated injection then f is an order embedding. Proof Since f is monotone it suffices to show that x 1 x 2 f(x 1 ) f(x 2 ) (7) for all x 1, x 2 X. If f(x 1 ) f(x 2 ) then x 1 (f f)(x 2 ) by (5). A residuated function always satisfies f f f = f, so since f is injective we have f f = id X. Hence x 1 (f f)(x 2 ) = x 2 as required. The most obvious examples of residuated functions are order isomorphisms: if f is an order isomorphism then the residual of f is just f 1. We now consider some less trivial examples. Example 9 Let R be a commutative ring. A subgroup I of (R, +, 0) is called an ideal of R if Ib = {ab : a I} I for all b R. Let X be the poset of ideals of R (ordered by subset inclusion) and let I X be a fixed ideal of R. Define a function f : X X by f(j) = IJ, where IJ denotes the ideal generated by the set {ab : a I, b J}. We then have f(j) K J {b R : Ib K} (8) for all J, K X, and as such f is residuated with residual f : X X given by f (K) = {b R : Ib K} (9) for all K X. Example 10 Let H be a Hilbert space, let X be the poset of subspaces of H, let Y be the poset of closed subspaces of H (X and Y are both ordered by subset inclusion) and define f : X Y by f(m) = M. We then have f(m) N M N (10) for all N Y, and as such f is residuated with residual f : Y X given by f (N) = N. 5

6 Example 11 A Boolean algebra is (in particular 1 ) a lattice in which the join and meet operations distribute over one another and upon which a complement is defined, e.g., the powerset of a set. Let X be a Boolean algebra, fix x X and define f : X X by f(y) = x y. We then have f(y) x y x z (11) for all y, z X, and as such f is residuated with residual f : X X given by f (z) = x z = x z (12) for all z X. 5 Conclusion Figure 3 summarises (in the style of a Hasse diagram) the relationships between the various types of order-preserving function that we have discussed above. order isomorphism residuated surjection residuated injection residuated function order embedding monotone bijection monotone surjection monotone injection monotone function Figure 3: Order-preserving functions arranged by strength. 1 A Boolean algebra must also have a greatest element and a least element. 6

7 References [1] T. S. Blyth. Lattices and Ordered Algebraic Structures. Springer, London, [2] G. W. Leibniz. A study in the calculus of real addition (after 1690). English translation in [3, pp ]. [3] G. H. R. Parkinson. Leibniz: Logical Papers. Clarendon Press, Oxford,